Applications of a generalized Galerkin's method to non-linear oscillations of two-degree-of-freedom systems

Chen, G.

doi:10.1016/0022-460x(87)90451-2

Cited by 21 publications

(13 citation statements)

References 14 publications

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“…Since that time, many approximate analytical solving methods have been adopted taking into consideration the specific properties of Jacobi elliptic functions. This includes the following methods: the Lindstedt-Poincaré [39], the method of multiple-scales [40], variation of parameters (Krylov-Bogoliubov) [41][42][43], the harmonic balance method [44][45][46][47][48] and the Galerkin method [49][50][51].…”

Section: Approximate Response Of Perturbed Nonlinear Oscillators: Quamentioning

confidence: 99%

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

Kovacic

Cvetićanin

Zukovic

et al. 2016

Journal of Sound and Vibration

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Section: Approximate Response Of Perturbed Nonlinear Oscillators: Quamentioning

confidence: 99%

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

Kovacic

Cvetićanin

Zukovic

et al. 2016

Journal of Sound and Vibration

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“…-The perturbation method [11], the asymptotic method [12], the averaging method [13] and the extended Galerkin [14] method are four approximate analytical methods that requires an assumption of small parameters in applying the algorithm to the equation and convergence. However, in solving mechanical and dynamical problems [1][2][3][4][5][6], coefficients of nonlinear terms and coefficients of exciting forces are rather large.…”

Section: Introductionmentioning

confidence: 99%

“…[8][9][10][11][12][13][14] were able to find and assess the real-valued solutions of the Duffing equation. These findings are considered sufficient for linear differential equations since the linear combination of the real and imaginary components of the solutions is also a solution to this type of equations.…”

Section: Introductionmentioning

confidence: 99%

A coupling successive approximation method for solving Duffing equation and its application

Bich

2014

Vietnam J. Mech.

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Abstract. The paper proposes an algorithm to solve a general Duffing equation, in which a process of transforming the initial equation to a resulting equation is proposed, and then the coupling successive approximation method is applied to solve the resulting equation. By using this algorithm a special physical factor and complex-valued solutions to the general Duffing equation are revealed. The proposed algorithm does not use any assumption of small parameters in the equation solving. The coupling successive procedure provides an analytic approximated solution in both real-valued or complex-valued solution. The procedure also reveals a formula to evaluate the vibration frequency, ϕ, of the non-linear equation. Since the first approximation solution is in a closed-form, the chaos index of the general Duffing equation and the chaotic characteristics of solutions can be predicted. Some examples are used to illustrate the proposed method. In the case of chaotic solution, the Pointcaré conjecture is used for solution verification.

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“…The motion of nonlinear two-degree-of-freedom (TDOF) oscillation system has been widely investigated in the past few decades [1][2][3][4][5][6][7][8]. TDOF systems are important in engineering because many practical engineering components consist of coupled vibrating systems that can be modeled using two-degree-of-freedom systems such as elastic beams supported by two springs and vibration of a milling machine [9].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

Lai

Lim

2006

Nonlinear Dyn

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An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-offreedom mass-spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a twomass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton's method and harmonic balance method. New and accurate higherorder analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newtonharmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-offreedom mass-spring system are analyzed and verified with published result, exact solutions and numerical integration data.

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Applications of a generalized Galerkin's method to non-linear oscillations of two-degree-of-freedom systems

Cited by 21 publications

References 14 publications

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

A coupling successive approximation method for solving Duffing equation and its application

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

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